Suppose $\mu(\Omega)<+\infty$. Let $(f_n)_n\subset L^1(\Omega,\mu)$ such that $\int_{\Omega}|f_n|d\mu\leq K<\infty$ for all $n\geq1$. Prove this statements are equivalent:
- $\forall\,\varepsilon>0$, $\exists\,\delta>0$ : $\sup_n\int_Ef_n\,du<\varepsilon$ if $\mu(E)<\delta$.
- $\lim\limits_{m\to\infty}\sup\limits_n \mu(x\,:\,|f_n(x)|\geq m)=0$
I have no idea how to proceed. Any idea?
The equivalence is wrong, mainly because 1. depends only on the collection $\{f_n\}$ while 2. depends on the full sequence $(f_n)$.
Assume for instance that $f_n=f_1$ for every $n$, for some integrable function $f_1$. The condition on the integrals on $\Omega$ always holds. Condition 1. holds as well (this is a classical exercise, decomposing the integral into the parts $[|f_1|\lt C]$ and $[|f_1|\geqslant C]$, for some well chosen $C$). But condition 2. holds if and only if $\mu[|f_1|\geqslant m]=0$ for every $m$ large enough, that is, when $f_1$ is "$\mu$-essentially bounded", which of course is not guaranteed.